Formulation of a mixed pair-and-triple interaction model and its determinant-like invariant

Develop a mathematically consistent framework that simultaneously incorporates pairwise forces F_{i,j} obeying Newton’s third law and triple interactions F_{i,j,k} obeying the antisymmetry rule F_{i,j,k} + F_{j,k,i} + F_{k,i,j} = −(F_{j,i,k} + F_{i,k,j} + F_{k,j,i}), and derive the corresponding determinant-like map that characterizes nontrivial rescalings to equilibrium in this mixed-interaction model.

Background

Section 3 treats 2-equilibrium (pairwise interactions) and Section 4 treats 3-equilibrium (triple interactions) separately. From a physics perspective, systems often feature both interaction types simultaneously.

The authors highlight the lack of a clear mathematical setting and invariant to analyze such mixed interactions, indicating a gap between the independently developed theories for r = 2 and r = 3.